Is #f(x)= cos(x+(pi)/4) # increasing or decreasing at #x=pi/3 #?

1 Answer
Shwetank Mauria
Nov 5, 2016

#f(x)=cos(x+pi/4)# is decreasing at #x=pi/3#

Explanation:

Whether a function is increasing or decreasing at a point depends on the value of its derivative at that point.

If derivative is positive, function is increasing and if it is negative, function is decreasing.

As #f(x)=cos(x+pi/4)#, #f'(x)=-sin(x+pi/4)#.

and at #x=pi/3#, #f'(pi/3)=-sin(pi/3+pi/4)=-sin((7pi)/12)=-sin(pi-(5pi)/12)=-sin((5pi)/12)#.

As #sin((5pi)/12)>0# (#(5pi)/12# being acute angle), #f'(pi/3)<0#

Hence, #f(x)=cos(x+pi/4)# is decreasing at #x=pi/3#

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